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We discuss a general framework for the analytic Langlands correspondence overan arbitrary local field F introduced and studied in our worksarXiv:1908.09677, arXiv:2103.01509 and arXiv:2106.05243, in particularincluding non-split and twisted settings. Then we specialize to the archimedeancases (F=C and F=R) and give a (mostly conjectural) description of the spectrumof the Hecke operators in various cases in terms of opers satisfying suitablereality conditions, as predicted in part in arXiv:2103.01509, arXiv:2106.05243and arXiv:2107.01732. Finally, we apply the tools of the analytic Langlandscorrespondence over archimedean fields in genus zero to the Gaudin model andits generalizations, as well as their q-deformations.

A general framework for the analytic Langlands correspondence